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Ch. 3. the Third Law of Thermodynamics (Pdf)

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Ch. 3. the Third Law of Thermodynamics (Pdf) 3-1 CHAPTER 3 THE THIRD LAW OF THERMODYNAMICS1 In sharp contrast to the first two laws, the third law of thermodynamics can be characterized by diverse expression2, disputed descent, and questioned authority.3 Since first advanced by Nernst4 in 1906 as the Heat Theorem, its thermodynamic status has been controversial; its usefulness, however, is unquestioned. 3.1 THE HEAT THEOREM The Heat Theorem was first proposed as an empirical generalization based on the temperature dependence of the internal energy change, ∆U, and the Helmholtz free energy change, ∆A, for chemical reactions involving condensed phases. As the absolute temperature, T, approaches zero, ∆U and ∆A by definition become equal, but The Heat Theorem stated that d∆U/dT and d∆A/dT also approach zero. These derivatives are ∆Cv and -∆S respectively. The statement that ∆Cv equals zero would attract little attention today in view of the abundance of experimental and theoretical evidence showing that the heat capacities of condensed phases approach zero as zero absolute temperature is approached. However, even today the controversial and enigmatic aspect of The Heat Theorem is the equivalent statement 1 Most of this chapter is taken from B.G. Kyle, Chem. Eng. Ed., 28(3), 176 (1994). 2 For a sampling of expressions see E. M. Loebl, J. Chem. Educ., 37, 361 (1960). 3 For extreme positions see E. D. Eastman, Chem. Rev., 18, 257 (1936). 4 All of Nernst's work in this area is covered in W. Nernst, The New Heat Theorem; Dutton: New York, 1926. 3-2 lim ∆S = 0 (3-1) T → 0 In 1912 Nernst offered a proof that the unattainability of zero absolute temperature was dictated by the second law of thermodynamics and was able to show that Eq. (3-1) follows from the unattainability principle. The latter result seems undisputed, but Nernst was unable to convince his contemporaries of the thermodynamic grounding of the unattainability principle. Many years of low-temperature research have firmly established the credibility of the unattainability principle and as a result it has been proposed as the third law of thermodynamics. This proposal has the merit of having all three laws expressed in phenomenological language and, of course, it leads to the useful result stated in Eq. (3-1). As a matter of convenience, it is possible to express ∆S for a process under consideration in terms of entropies of formation of participating species because in such a calculation there is a cancellation of the entropies of the constituent elements. For this reason the entropy of an element may be assigned any value. According to Eq. (3-1), at zero absolute temperature the entropy changes for formation reactions will be zero and it is convenient to set elemental entropies equal to zero as recommended by Lewis and Randall.5 This results in the familiar statement that the entropy of every perfect crystalline substance can be taken as zero at zero absolute temperature and is, of course, the convention employed in the determination of "absolute" entropies. 3.2 CONFORMANCES, EXCEPTIONS, AND INTERPRETATIONS Undoubtedly the most convincing confirmation of the Heat Theorem involved the calculation of absolute entropies from calorimetric measurements on pure substances which were then used to calculate entropy changes for chemical reactions. These calculated values were in agreement with entropy changes determined from the temperature dependence of experimentally measured equilibrium constants.6 Later, it was shown through the use of 5 G. N. Lewis and M. Randall, Thermodynamics and the Free Energy of Chemical Substances, McGraw-Hill, New York, 1923, Chap. 31. 6 Other successful applications of the third law are given by K. Denbigh, The Principles of Chemical Equilibrium, 3rd ed., Cambridge University Press, Cambridge, 1971, Chap. 13. 3-3 quantum statistical mechanics that spectroscopic data could be used to calculate absolute entropies in excellent agreement with those calculated from calorimetric data.7 Quantum statistical mechanics also provides the microscopic interpretation of zero entropy for a perfect crystal as well as quantitative corrections for those few errant substances exhibiting small positive entropy values at zero absolute temperature. The statement that the lowest energy state of the crystal is nondegenerate (only a single quantum state is available to it) is easily visualized as a perfectly ordered crystal where only a single arrangement of atoms, molecules, or ions on the crystal lattice is possible. Thus, in terms of Boltzmann's famous equation S = k ln Ω (3-2) it may be stated that for Ωo = 1 at T = 0 the entropy, So, is zero. Exceptions to So equal to zero are explained in terms of "frozen-in" disorder. For example, a linear molecule such as carbon monoxide can take two possible orientations on a lattice site, CO or OC. Orientations on adjacent sites such as COOC or OCCO represent a slightly higher energy state than ordered orientations such as COCO and are therefore favored at higher temperatures. While there is a tendency for the crystal to move toward the low-energy, ordered state on cooling, the rate at which molecular orientations proceed slows to a standstill and a state of "frozen-in" disorder results at zero absolute temperature. If the orientations of the CO molecule were completely random, there would be 2N possible configurations on a lattice of N sites (two possibilities per site). N Setting Ωo = 2 in Eq. (3-2) leads to So = Rln2 which is also seen to be the entropy change on forming an equimolar binary mixture.8 The value of Rln2 is extremely close to the observed difference between calorimetric and spectroscopic absolute entropies. The vast majority of substances conform to So equal to zero and can be visualized as forming crystals of perfect order (Ωo = 1). The few exceptions can be explained in terms of "frozen-in" disorder in a manner similar to that described for carbon monoxide. Here there is seen to be a close correspondence between 7 See Sec. 3-6. 8 See Eq. (14-22) of this textbook. 3-4 entropy and disorder in a spatial sense. Unfortunately, there are other systems conforming to the Heat Theorem that place a strain on this interpretation. We now examine these systems. Measurements of phase equilibrium data for pure substances show that the slope of the solid-vapor coexistence curve for many substances and the slope of the liquid-vapor coexistence curves for 4He and 3He approach zero as zero absolute temperature is approached.9 From the Clapeyron equation, dP ∆S dT ∆v (3-3) and the fact that ∆v is finite, we obtain the result that ∆S = 0 and conclude that Eq. (3-1) applies to these phase changes. Both helium isotopes remain liquid under their own vapor pressure down to zero Kelvin and both require a pressure considerably higher than their vapor pressures in order to form a solid phase and the appropriate calculations show10 that Eq. (3-1) also applies to the solid- liquid phase transition. Thus, if the Lewis and Randall convention is used, the entropies of pure liquids and vapors also have zero entropies. While it may be possible to argue that these systems are nondegenerate in their lowest energy state, the simple picture of zero entropy corresponding to perfect order does not seem appropriate, at least in a physical sense. The interpretation is further strained when the behavior of glasses in the low-temperature limit is considered. The Maxwell relationship ∂S ∂V = - (3-4) ∂ ∂ P T T P together with Eq. (3-1) leads to ∂V lim = 0 (3-5) T → 0 ∂ T P Thermal coefficients of expansion for many substances have been measured at 9 J. A. Beattie and I. Oppenheim, Principles of Thermodynamics; Elsevier, Amsterdam, 1979, Chap. 11. 10 ibid. 3-5 temperatures approaching absolute zero. As expected, Eq. (3-5) is obeyed by crystalline solids, but one may be surprised to learn that it is also obeyed by glasses.11 Here a microscopic physical interpretation hardly seems possible. Systems comprised of liquids, vapors, and glasses strain to the breaking point the putative association of zero entropy with perfect spatial order. These are the systems that prompt us to ask Is there a microscopic physical interpretation of the Heat Theorem applicable to all systems? One could argue that the association of entropy with spatial order is naive and that Ω0 = 1 only means that the system is nondegenerate. For example, both Fermi-Dirac and Bose-Einstein gases have been shown to be nondegenerate12 and therefore have So = 0. In the case of crystalline solids Ωo = 1 can be interpreted physically as spatial order, but no such microscopic description of the gases in physical terms is possible. Instead Ωo can only be seen as a logical construct that allows a mathematical treatment of the system. The answer to the question is No! only an explanation in logical terms is possible. This is yet another instance of our inability to obtain a microscopic view of entropy in anything other than logical terms. If there is no physical microscopic interpretation of the Heat Theorem, then what is the basis for its existence? As will be shown below, the answer is that Eq. (3-1) is dictated by the logical structure of thermodynamics. 3.3 THE CLASSICAL THERMODYNAMIC VIEW The absolute temperature scale is defined in terms of the performance of a Carnot engine T 1 |Q | = 1 (3-6) T 2 |Q2 | where Q2 is the input heat at T2 and Q1 is rejected heat at T1. Instrumental in the derivation of Eq. (3-6) is a second-law statement such as It is impossible to 11 G. K. White, Cryogenics, 4, 2 (1964).
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